Hilbert’s Problems

by étienne ghys

R. Mohan

This article is a translation of Les problèmes de Hilbert, written in 2010 by Étienne Ghys for the French mathematics magazine Images des Mathématiques, and is published here with permission.

In August 1900, David Hilbert delivered a lecture which remains popular among mathematicians. Entitled “Mathematical Problems”, it presented a list of twenty-three open problems that Hilbert considered as important for the future of mathematics. To discuss all of these problems, and review their current status one hundred and ten years later, would require twenty-three separate articles. Here, I will only quote a few excerpts from the introduction to this lecture.

First, what is a good problem according to Hilbert?

Who amongst us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

The role of problems in the development of mathematics

The deep significance of certain problems for the advancement of the mathematical sciences in general, and the important role that they play in the work of the individual investigator, are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research pursues certain problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon.

A good problem must be crystal clear

An old French mathematician said: “A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man you meet on the street.” This clearness and ease of comprehension, insisted on here for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehensible attracts us, whereas the complicated repels us.

Hilbert alludes to a comment by Joseph Diaz Gergonne, the French geometer who, in 1810, founded the first real mathematical journal: the Annales de mathématiques pures et appliquées, also known as Annales de Gergonne.1 Anyway, we should meditate on this sentence sometimes: “What is clear and easily comprehended attracts,whereas the complicated repels us.”

A problem must be difficult, but approachable

I think everyone will agree with the following comment: if it is too easy, a problem is not interesting, but if it is too difficult, it is useless. How does one know in advance, though, that a problem is not inaccessible?

A mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide post on the maze-like paths to hidden truths, and ultimately serve as a reminder of our pleasure on its successful solution.

Where do good problems come from?

Non-mathematicians often think that mathematics should be applied and should be useful to solve concrete problems. In front of problems of “pure mathematics”, they ask: “what is it useful for?” Hilbert explains that things are more subtle.

The first problems came from the real world.

Having just recalled the general importance of problems in mathematics, let us enquire from what sources this science derives its problems. Surely the first and oldest problems in every branch of mathematics spring from experience and are inspired from the world of external phenomena.

However, these concrete problems generate new problems within mathematics, which a priori do not promise any direct “utility”.

But, in the furtherance of a branch of mathematics encouraged by the success of its solutions, the human mind becomes conscious of its free spirit. It evolves on its own, often without appreciable influence from reality, by a combination of logic, generalization and specialization, by separating and collecting ideas in fortuitous ways, thereof giving rise to even more new and fruitful problems, and thus appearing itself as the real questioner.

These pure problems, sometimes, reflect on the concrete in return.

In the meantime, while the creative power of pure reason is at work, the outer world again comes into play, and forces upon us new questions from actual experience that open up newer branches of mathematics. While we seek to conquer these new fields of knowledge from the realm of pure thought, we often find answers to old unsolved problems, and thus at the same time advance successfully the old theories. It seems to me that the numerous and surprising analogies, and that apparently prearranged harmony which the mathematician so often perceives in the questions, methods and ideas of the various branches of his science, have their origin in this ever-recurring interplay between thought and experience.

Rigour before everything!

Today, the following message can hardly be read in national curricula—

Indeed the requirement of rigour, which has become proverbial in mathematics, corresponds to a universal philosophical necessity of our understanding; and, on the other hand, only by satisfying this requirement do the thought content and the suggestiveness of the problem attain their full effect. A new problem, especially when it comes from the world of outer experience, is like a young twig, which thrives and bears fruit only when it is grafted carefully and in accordance with strict horticultural rules upon the old stem, the established achievements of our mathematical science.

Besides, it is an error to believe that rigour in the proof is the enemy of simplicity. On the contrary we find it confirmed by numerous examples that the rigorous method is at the same time the simpler and the more easily comprehended. The very effort for rigour forces us to find out simpler methods of proof. It also frequently leads the way to methods which are more capable of development than the old methods of less rigour.

Generalise to understand better but understand particular cases

Does one go from the general to the particular? Or from the particular to the general? These are two different but complementary approaches and are two ways to “do math”.

If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears only as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the same time we come to possess a method which is applicable also to related problems.

In dealing with certain mathematical problems, specialization plays, as I believe, a still more important part than generalization. Perhaps in most cases where we seek in vain the answer to a question, the cause of the failure lies in the fact that problems which are simpler and easier than the one in hand have themselves been either not at all, or incompletely solved. All depends, then, on finding these easier problems,and on solving them by means of devices as perfect as possible, and also via concepts capable of generalization. This rule is one of the most important levers for overcoming mathematical difficulties, and it seems to me that it is used almost always, though perhaps unconsciously.

This reminds me of this beautiful quotation of Grothendieck, who explains in Récoltes et semailles2 his way of solving problems:

Let us, for instance, consider the task of proving a so-far hypothetical theorem (which, for some people, mathematical work seems to reduce to). I can see two extreme approaches to tackle it. One is that of hammer and chisel, when the problem at hand is considered as a big, smooth, hard nut, the interior of which is wanted—the nourishing flesh protected by the shell. The principle is simple: one sets the edge of the chisel against the shell, and one hits hard. If necessary, one hits again at several different places, until the shell breaks up—and one is satisfied. … I could picture the second approach, still with the nut image one wants to crack. The first parable that came to my mind is that one plunges the nut into an emollient liquid, just water maybe, why not, and one scrubs from time to time, and for the rest, one leaves it to time. The shell softens with the weeks and the months, and when time is mature, simple hand pressure is enough–-the shell opens up as that of a ripe avocado. Alternately, one leaves the nut under the sun and under the rain, and maybe under the winter frosts. When time is mature, what drills the shell is a delicate shoot out of the substantial flesh, as if playing—or, to say better, the shell will have opened up by itself to let it pass through.… The reader who is the least familiar with some of my works will find no difficulty in recognising which of these two approaches is `mine’.

Sometimes, one can show that a solution to a problem is impossible

For centuries, there were attempts to square the circle, until it was proved to be impossible towards the end of the nineteenth century.

Occasionally it happens that we seek the solution under insufficient hypotheses or in an incorrect sense, and for this reason do not succeed. The problem then arises: to show the impossibility of the solution under the given hypotheses, or in the sense contemplated. Such proofs of impossibility were effected by the ancients, for instance when they showed that the ratio of the hypotenuse to the side of an isosceles right-triangle is irrational. In later mathematics, the question as to the impossibility of certain solutions plays a prominent part.

The “conviction” of the mathematician

It is probably this important fact along with other philosophical reasons that gives rise to the conviction (which every mathematician shares, but which no one has as yet supported with a proof) that every definite mathematical problem must necessarily be susceptible to an exact settlement, either in the form of an actual answer to the question asked; or by the proof of the impossibility of its solution, and therewith the necessary failure of all attempts. …

This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is this problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus.

Thirty years later, Hilbert was forced to revise his “conviction which every mathematician shares, but which no one till then had yet shown, by providing a proof”. Gödel’s theorem, in 1931, would indeed show that in every formal system containing arithmetic, there exists statements which are impossible to prove and to refute, with their negation being equally impossible to prove as well. This is liable to test our “convictions”. This was a deep change in our vision of right and wrong. However, it must be said that even though, nearly all mathematicians are aware of this theorem, they do not believe it too much in their everyday practice, and they stick to the conviction mentioned by Hilbert. Sometimes, we have convictions which we also know, deep inside, are not completely justifiable…

On 8 September 1930, a year before Gödel’s theorem, Hilbert was lecturing about “The knowledge of nature and logic”. The conclusion of his lecture was broadcast on the radio, where Hilbert claimed:

We must not believe those, who today with philosophical bearing and a tone of superiority, prophesy the downfall of culture and accept the ignorabimus. For us there is no ignorabimus, and in my opinion even none whatever in natural science. In place of the foolish ignorabimus let our slogan stand:

We must know,
We will know.

This beautiful conclusion, “Wir müssen wissen, Wir werden wissen”, was engraved on Hilbert’s grave (he died in 1943), which could imply that he upheld it all his life.

An optimistic conclusion

The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved, numerous others arise in its place.

Footnotes

In an article, (page 88, footnote), Quételet quotes a letter from Gergonne: “For a long time I have repeated to my students that we do not have the final say about a problem in science as long as one cannot bring it to the point where it can be told to a man met by chance in the street.” Thanks to Karine Chemla for the reference, which is: des lettres et des beaux-arts de Belgique. Académie royale des sciences. Nouveaux Mémoires de L’Académie Royale des Sciences et Belles-Lettres de Bruxelles. Volume t.4. 1820. ↩